Questions C4 (1162 questions)

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OCR C4 Q7
7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4
1
3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3
6
1 \end{array} \right)\) respectively.
  1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3
    - 7
    9 \end{array} \right) + t \left( \begin{array} { c } 2
    - 3
    1 \end{array} \right)$$
  2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
  3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
OCR C4 Q9
9 \end{array} \right) + t \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (ii) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(iii) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
8. \(f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\).
(i) Express \(\mathrm { f } ( x )\) in partial fractions.
(ii) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(iii) State the set of values of \(x\) for which your expansion is valid.
OCR C4 Q1
  1. Evaluate
$$\int _ { 0 } ^ { \pi } \sin x ( 1 + \cos x ) d x$$
OCR C4 Q2
  1. (i) Simplify
$$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$ (ii) Express $$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 }$$ as a single fraction in its simplest form.
OCR C4 Q3
3. Find the exact value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x d x$$
OCR C4 Q4
4.
\includegraphics[max width=\textwidth, alt={}]{23bd8979-9ba6-4e77-a3d1-88feb5e5a5b3-1_444_728_1425_536}
The diagram shows the curve with parametric equations $$x = t + \sin t , \quad y = \sin t , \quad 0 \leq t \leq \pi$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q5
5. Given that \(y = - 2\) when \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR C4 Q6
6. (i) Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
(ii) Using the substitution \(u = x ^ { 2 } + 4\), evaluate $$\int _ { 0 } ^ { 2 } \frac { 5 x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$$
OCR C4 Q7
  1. A curve has the equation
$$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point \(P\) on the curve has coordinates \(( - 1,3 )\).
  1. Show that the normal to the curve at \(P\) has the equation \(y = 2 - x\).
  2. Find the coordinates of the point where the normal to the curve at \(P\) meets the curve again.
OCR C4 Q8
8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
  2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
  3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).
OCR C4 Q9
9. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where \(k\) is an integer to be found.
  3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C4 Q1
1. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 }$$ Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
OCR C4 Q2
  1. A curve has the equation
$$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).
OCR C4 Q3
3. Find
  1. \(\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x\),
  2. \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).
OCR C4 Q4
4.
\includegraphics[max width=\textwidth, alt={}, center]{c7b867af-0730-459e-9c76-15eb07b9e476-1_465_976_1539_388} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$
  1. Find a cartesian equation for the curve. The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\).
  2. Using integration, with the substitution \(x = \tan u\), find the area of the shaded region.
OCR C4 Q5
5. (i) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.
(iii) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.
OCR C4 Q6
6. (i) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(ii) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(iii) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).
OCR C4 Q7
7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
  2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
  4. Find the position vector of \(C\).
OCR C4 Q8
8. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,
  1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
  2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.
OCR C4 Q1
  1. \(\mathrm { f } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 }\).
Find the values of the constants \(A , B , C\) and \(D\) such that $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$
OCR C4 Q2
  1. Use the substitution \(u = 1 - x ^ { \frac { 1 } { 2 } }\) to find
$$\int \frac { 1 } { 1 - x ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$$
OCR C4 Q3
  1. A curve has the equation
$$4 \cos x + 2 \sin y = 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
  2. Find an equation for the tangent to the curve at the point ( \(\frac { \pi } { 3 } , \frac { \pi } { 6 }\) ), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.
OCR C4 Q4
4. (i) Express \(\frac { 3 x + 6 } { 3 x - x ^ { 2 } }\) in partial fractions.
(ii) Evaluate \(\int _ { 1 } ^ { 2 } \frac { 3 x + 6 } { 3 x - x ^ { 2 } } \mathrm {~d} x\).
OCR C4 Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{825f6c7d-5399-4e7f-bacd-b7c0831aab06-1_408_858_1893_488} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\) is rotated through four right angles about the \(x\)-axis. Find, in terms of \(\pi\) and e, the exact volume of the solid formed.
OCR C4 Q6
6. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$
  1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
  2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
  4. Find, to 1 significant figure, the percentage error in your answer to part (c).